The first thing that came to mind was Fia...it's an Irish name, and I once knew an Eastern European girl who had that name. I think it may have been short for Zofia? That might be a good option for your family that includes both heritages.

Elena comes to mind as a really pretty Russian name with less of a dramatic flair than Anastasia & Tatiana. I think it wears well on a girl of any nationality in 2013. Fiona is a pretty Irish name that is recognizable in the states that also feels very 2013 for me. Both names would be easily pronounced by the other culture because of the similar sound structure. So if you want an Irish name that Dida could easily pronounce, this might be a good choice.

I also think that Lana & Nina & Anya are sweet. I think there's an Irish spelling of Anya that shares the Russian pronunciation, but I'm not sure what it is. Good luck!